# Definite integral with Bessel-J function

I try to find an closed form answer the following integral, $$\int_{0}^{\infty}dx\,x^{m}J_l(Qx)e^{-x^2},\quad m\in\mathbb{Z}, l\in\mathbb{Z},Q>0$$ where $$J_l(Qx)$$ is Bessel function of the first kind, but I am not sure that it exists. Can anybody help me?

Wolfram Mathematica says that the answer (with assumption $$\mathrm{Re}\,(l+m)>-1$$) is $$2^{-l-1}Q^l\Gamma\left(\frac{m+l+1}{2}\right){}_1\tilde{F}_1\left(\frac{m+l+1}{2},l+1,-\frac{Q^2}{4}\right),$$ where $${}_1\tilde{F}_1$$ is regularized confluent hypergeometric function. However, I feel that due to $$m,l\in\mathbb{Z}$$ more simple answer can exist.

• please define $J_l$ Mar 19 '20 at 20:25
• @phdmba7of12 I have edited the question Mar 19 '20 at 20:29
• not sure if this relationship would help at all (found on another post here) $$e^{-\frac{x^2}{4}}=\lim_{\nu\rightarrow\infty}\Gamma(\nu+1)\left(\frac{2}{\sqrt\nu x}\right)^\nu J_\nu\left(\sqrt{\nu} x\right),$$ Mar 19 '20 at 20:35
• @phdmba7of12 to be honest, I do not understand how it can help. You suggest to plug this representation of exp into the integral? Mar 19 '20 at 20:36
• It can be found in Gradshteyn and Ryzhik as formula number 6.631.1. Mar 19 '20 at 20:47